This chapter will teach you how one can cluster and classify datasets. There is different methods to perform the clustering of the data, but in a very simple point of view you cluster what is similar. For that you can use different parameters, from simple concepts of means to more complex concepts like data populations distances.
The data set Boston is included in the library {MASS} and refers to the “Housing Values in Suburbs of Boston”. The data compiled in the dataset orginates from the following papers: Harrison and Rubinfeld (1978) & Belsley et al 1980
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The Boston data set includes the Housing Values in Suburbs of Boston (from 1978 - 1980). The information included in this data set includes values regarding Boston city crime, nitrogen oxide concentrations, average rooms per dwelling, lower status of the population, pupil-teacher ratio in town, etc..
## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
## [1] 506 14
You can see that the data includes a total of 506 observations distributed into the following 14 variables: - crim = per capita crime rate by town. - zn = proportion of residential land zoned for lots over 25,000 sq.ft. - indus = proportion of non-retail business acres per town. - chas = Charles River dummy variable (= 1 if tract bounds river; 0 otherwise). - nox = nitrogen oxides concentration (parts per 10 million). - rm = average number of rooms per dwelling. - age = proportion of owner-occupied units built prior to 1940. - dis = weighted mean of distances to five Boston employment centres. - rad = index of accessibility to radial highways. - tax = full-value property-tax rate per $10,000. - ptratio = pupil-teacher ratio by town. - black = 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town. - lstat = lower status of the population (percent). - medv = median value of owner-occupied homes in $1000s.
## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00
## corrplot 0.84 loaded
## crim zn indus chas nox rm age dis rad tax
## crim 1.00 -0.20 0.41 -0.06 0.42 -0.22 0.35 -0.38 0.63 0.58
## zn -0.20 1.00 -0.53 -0.04 -0.52 0.31 -0.57 0.66 -0.31 -0.31
## indus 0.41 -0.53 1.00 0.06 0.76 -0.39 0.64 -0.71 0.60 0.72
## chas -0.06 -0.04 0.06 1.00 0.09 0.09 0.09 -0.10 -0.01 -0.04
## nox 0.42 -0.52 0.76 0.09 1.00 -0.30 0.73 -0.77 0.61 0.67
## rm -0.22 0.31 -0.39 0.09 -0.30 1.00 -0.24 0.21 -0.21 -0.29
## age 0.35 -0.57 0.64 0.09 0.73 -0.24 1.00 -0.75 0.46 0.51
## dis -0.38 0.66 -0.71 -0.10 -0.77 0.21 -0.75 1.00 -0.49 -0.53
## rad 0.63 -0.31 0.60 -0.01 0.61 -0.21 0.46 -0.49 1.00 0.91
## tax 0.58 -0.31 0.72 -0.04 0.67 -0.29 0.51 -0.53 0.91 1.00
## ptratio 0.29 -0.39 0.38 -0.12 0.19 -0.36 0.26 -0.23 0.46 0.46
## black -0.39 0.18 -0.36 0.05 -0.38 0.13 -0.27 0.29 -0.44 -0.44
## lstat 0.46 -0.41 0.60 -0.05 0.59 -0.61 0.60 -0.50 0.49 0.54
## medv -0.39 0.36 -0.48 0.18 -0.43 0.70 -0.38 0.25 -0.38 -0.47
## ptratio black lstat medv
## crim 0.29 -0.39 0.46 -0.39
## zn -0.39 0.18 -0.41 0.36
## indus 0.38 -0.36 0.60 -0.48
## chas -0.12 0.05 -0.05 0.18
## nox 0.19 -0.38 0.59 -0.43
## rm -0.36 0.13 -0.61 0.70
## age 0.26 -0.27 0.60 -0.38
## dis -0.23 0.29 -0.50 0.25
## rad 0.46 -0.44 0.49 -0.38
## tax 0.46 -0.44 0.54 -0.47
## ptratio 1.00 -0.18 0.37 -0.51
## black -0.18 1.00 -0.37 0.33
## lstat 0.37 -0.37 1.00 -0.74
## medv -0.51 0.33 -0.74 1.00
By drawing the correlation plot you can see some variables are highly positive or negatively correlated with each other.
High positive numbers (Blue) mean that variables value will shift together in the same direction. For example, when you look at the crime data, you can see its higly positvely correlated with the tax, rad and lstat variables. Meaning that the higher the index of acessibility to radial highways, the higher value of property-tax rate per $10.000, and the amount of low status of the population the bigger amount of crime you can witness on those areas.
On the other side, you can also observe some High negative numbers (Red) meaning that while the value of one variable increases the other one will decrease. For example, you can see how lstat (percentage of the lower status of the population) inversely correlates with medv (median value of owner-occupied homes in $1000s) and rm (average number of rooms per dwelling). Meaning that the higher amount of low status population the less amount of rooms per dwelling, and the less amount of owner occupied homes with the value of $1000.
## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865
## [1] "matrix"
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.419367 -0.410563 -0.390280 0.000000 0.007389 9.924110
## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610
## crime
## low med_low med_high high
## 127 126 126 127
## # A tibble: 506 x 14
## # Groups: crime [4]
## zn indus chas nox rm age
## * <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.28454827 -1.2866362 -0.2723291 -0.1440749 0.4132629 -0.11989477
## 2 -0.48724019 -0.5927944 -0.2723291 -0.7395304 0.1940824 0.36680343
## 3 -0.48724019 -0.5927944 -0.2723291 -0.7395304 1.2814456 -0.26554897
## 4 -0.48724019 -1.3055857 -0.2723291 -0.8344581 1.0152978 -0.80908783
## 5 -0.48724019 -1.3055857 -0.2723291 -0.8344581 1.2273620 -0.51067434
## 6 -0.48724019 -1.3055857 -0.2723291 -0.8344581 0.2068916 -0.35080997
## 7 0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.3880270 -0.07015919
## 8 0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.1603069 0.97784057
## 9 0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.9302853 1.11638970
## 10 0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.3994130 0.61548134
## # ... with 496 more rows, and 8 more variables: dis <dbl>, rad <dbl>,
## # tax <dbl>, ptratio <dbl>, black <dbl>, lstat <dbl>, medv <dbl>,
## # crime <fctr>
## [1] 506
## zn indus chas nox rm
## 223 -0.48724019 -0.71961001 3.6647712 -0.41159834 0.845930986
## 419 -0.48724019 1.01499462 -0.2723291 1.07272553 -0.466305712
## 433 -0.48724019 1.01499462 -0.2723291 0.25289548 0.199775388
## 240 0.79907391 -0.90473168 -0.2723291 -1.09335175 0.457383677
## 459 -0.48724019 1.01499462 -0.2723291 1.36613839 0.023292362
## 303 0.97058245 -0.73564417 -0.2723291 -1.05020280 0.299402903
## 61 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.773727758
## 212 -0.48724019 -0.07970124 3.6647712 -0.56693456 -1.253363081
## 352 2.08538800 -1.37701059 -0.2723291 -1.24005818 0.418955921
## 376 -0.48724019 1.01499462 -0.2723291 1.00368721 1.463621579
## 280 0.37030254 -1.13795583 -0.2723291 -0.96476788 0.750573221
## 56 3.37170210 -1.44552019 -0.2723291 -1.30909650 1.372533565
## 189 1.44223095 -1.12192167 -0.2723291 -1.01568364 0.386221166
## 262 0.37030254 -1.04466617 -0.2723291 0.79657225 1.758234373
## 294 -0.48724019 0.40569652 -0.2723291 -1.01568364 -0.224353176
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## 276 1.22784527 -0.68899934 -0.2723291 -0.92938574 0.810349730
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## 496 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.874778524
## 32 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.302631937
## 398 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.765188257
## 46 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.857699521
## 372 -0.48724019 1.01499462 -0.2723291 0.65849561 -0.097683907
## 234 -0.48724019 -0.71961001 -0.2723291 -0.41159834 2.792937279
## 400 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.615746985
## 80 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.584435480
## 493 -0.48724019 2.42017014 -0.2723291 0.46864023 -0.429301206
## 418 -0.48724019 1.01499462 -0.2723291 1.07272553 -1.395688102
## 98 -0.48724019 -1.20209248 -0.2723291 -0.94664532 2.539598741
## 337 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.591551731
## 443 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.093414156
## 245 0.45605682 -0.76917014 -0.2723291 -1.06746238 -0.984368790
## 88 -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.232892677
## 302 0.97058245 -0.73564417 -0.2723291 -1.05020280 0.434611674
## 70 0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.568779727
## 429 -0.48724019 1.01499462 -0.2723291 1.07272553 -0.130418661
## 489 -0.48724019 2.42017014 -0.2723291 0.46864023 -1.182200570
## 121 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.590128481
## 14 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.477691714
## 209 -0.48724019 -0.07970124 3.6647712 -0.56693456 -0.314017939
## 161 -0.48724019 1.23072696 3.6647712 0.43412107 -0.049293399
## 431 -0.48724019 1.01499462 -0.2723291 0.25289548 0.090185122
## 9 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.930285282
## 473 -0.48724019 1.01499462 -0.2723291 0.21837632 0.216854391
## 179 -0.48724019 -1.03300497 -0.2723291 -0.38570897 0.818889232
## 4 -0.48724019 -1.30558569 -0.2723291 -0.83445805 1.015297761
## 54 0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.407952453
## 318 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -0.715374500
## 386 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.434115858
## 16 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.641365488
## 404 -0.48724019 1.01499462 -0.2723291 1.19354259 -1.331641842
## 281 0.37030254 -1.13795583 -0.2723291 -0.96476788 2.185209437
## 79 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.074911903
## 175 -0.48724019 -1.03300497 -0.2723291 -0.38570897 -0.605784233
## 250 0.45605682 -0.76917014 -0.2723291 -1.06746238 0.616787701
## 185 -0.48724019 -1.26477147 -0.2723291 -0.57556435 -0.968713038
## 15 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 408 -0.48724019 1.01499462 -0.2723291 0.90012973 -0.963020037
## 3 -0.48724019 -0.59279438 -0.2723291 -0.73953036 1.281445551
## 345 1.87100232 -1.07236154 -0.2723291 -0.61008351 0.838814735
## 93 0.71331963 0.56895343 -0.2723291 -0.78267931 0.223970642
## 326 -0.48724019 -0.54760720 -0.2723291 -0.53241540 0.208314890
## 55 2.72854505 -1.04029322 -0.2723291 -1.24868797 -0.564509977
## 233 -0.48724019 -0.71961001 -0.2723291 -0.41159834 2.921029798
## 23 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.203004422
## 183 -0.48724019 -1.26477147 -0.2723291 -0.57556435 1.238748045
## 307 0.92770532 -1.30558569 -0.2723291 -0.71364099 1.615909352
## 336 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.352445695
## age dis rad tax ptratio
## 223 0.324172928 -0.248345051 -0.1779443 -0.60068166 -0.48755665
## 419 1.116389695 -0.946209387 1.6596029 1.52941294 0.80577843
## 433 0.221149222 -0.757294495 1.6596029 1.52941294 0.80577843
## 240 -0.936979326 1.137315760 -0.4076377 -0.64221553 -0.85708096
## 459 0.537325421 -0.480570747 1.6596029 1.52941294 0.80577843
## 303 -1.782484217 0.805741156 -0.2927910 -0.47014661 -1.08803366
## 61 -0.084369352 1.629073854 -0.1779443 -0.73715011 0.57482574
## 212 0.711399958 -0.061757165 -0.6373311 -0.77868399 0.06672981
## 352 -1.160789444 3.284049986 -0.6373311 0.01639310 -0.07184181
## 376 1.041786323 -1.177152855 1.6596029 1.52941294 0.80577843
## 280 -1.292233482 0.145156405 -0.5224844 -1.14062207 -1.64232012
## 56 -1.658145262 2.327745519 -0.5224844 -1.08128796 -0.25660396
## 189 -1.402362270 0.366459420 -0.5224844 -0.06074124 -1.50374851
## 262 0.739820290 -0.786073385 -0.5224844 -0.85581834 -2.51994036
## 294 -1.782484217 0.810965047 -0.6373311 -0.70748306 -1.13422420
## 485 -0.947636951 -0.033738114 1.6596029 1.52941294 0.80577843
## 241 -0.507121797 1.206746019 -0.4076377 -0.64221553 -0.85708096
## 210 1.116389695 0.037971662 -0.6373311 -0.77868399 0.06672981
## 25 0.906789743 0.287103768 -0.6373311 -0.60068166 1.17530274
## 492 1.073759197 -0.915103491 -0.6373311 1.79641644 0.75958789
## 172 1.020471073 -0.667870981 -0.5224844 -0.03107419 -1.73470120
## 87 -0.833955621 0.300210985 -0.7521778 -0.95668632 0.02053927
## 244 -2.159053622 1.538890501 -0.4076377 -0.64221553 -0.85708096
## 276 -0.915664077 0.224274608 -0.6373311 -0.91515245 -0.39517558
## 277 -0.695406500 0.471174688 -0.6373311 -0.91515245 -0.39517558
## 389 1.116389695 -1.047410401 1.6596029 1.52941294 0.80577843
## 410 1.116389695 -1.106297898 1.6596029 1.52941294 0.80577843
## 182 -0.226471014 -0.568522074 -0.7521778 -1.27709053 -0.30279450
## 300 -2.080897708 1.915153117 -0.5224844 -0.29807769 -1.68851066
## 206 -1.643935096 0.071404563 -0.6373311 -0.77868399 0.06672981
## 211 0.835738912 0.038968950 -0.6373311 -0.77868399 0.06672981
## 288 -1.324206356 1.672669599 -0.4076377 -0.68374941 -0.85708096
## 367 0.810871121 -0.970096815 1.6596029 1.52941294 0.80577843
## 95 0.309962761 -0.085502124 -0.6373311 -0.82021787 -0.11803234
## 220 0.846396537 -0.205034246 -0.5224844 -0.78461740 -0.94946204
## 349 -1.381047021 2.160296071 -0.6373311 -0.76088376 -0.67231881
## 353 -1.778931675 3.284049986 -0.6373311 0.01639310 -0.07184181
## 266 -0.205155765 -0.858875429 -0.5224844 -0.85581834 -2.51994036
## 361 0.690084708 -0.606371538 1.6596029 1.52941294 0.80577843
## 270 -0.251338805 0.058154876 -0.7521778 -1.09908819 0.06672981
## 396 1.073759197 -0.982729133 1.6596029 1.52941294 0.80577843
## 381 0.828633829 -1.129567958 1.6596029 1.52941294 0.80577843
## 292 -1.452097852 0.627653966 -0.6373311 -0.96855315 0.34387304
## 81 -1.246050442 0.762525331 -0.6373311 -0.75495035 0.25149196
## 328 -0.883691203 0.769743799 -0.5224844 -0.71934988 0.52863520
## 374 1.116389695 -1.244636028 1.6596029 1.52941294 0.80577843
## 460 0.562193212 -0.511724132 1.6596029 1.52941294 0.80577843
## 216 -0.929874243 0.071404563 -0.6373311 -0.77868399 0.06672981
## 165 0.825081288 -0.652056838 -0.5224844 -0.03107419 -1.73470120
## 382 1.084416821 -1.080795813 1.6596029 1.52941294 0.80577843
## 264 0.920999910 -0.815042235 -0.5224844 -0.85581834 -2.51994036
## 426 0.952972784 -0.895395175 1.6596029 1.52941294 0.80577843
## 66 -1.803799466 1.337533251 -0.6373311 -0.42267932 -1.08803366
## 255 -1.299338565 2.576450217 -0.9818712 -0.55321437 -0.94946204
## 36 -0.013318520 -0.206458943 -0.5224844 -0.76681717 0.34387304
## 153 0.690084708 -1.037579988 -0.5224844 -0.03107419 -1.73470120
## 128 0.974288033 -0.953000445 -0.6373311 0.17066179 1.26768382
## 53 -1.686565595 1.434032764 -0.6373311 -0.98041997 -0.76469989
## 228 0.402328842 -0.275129364 -0.1779443 -0.60068166 -0.48755665
## 13 -1.050660656 0.786365270 -0.5224844 -0.57694801 -1.50374851
## 470 -0.421860800 -0.461289840 1.6596029 1.52941294 0.80577843
## 425 0.071942477 -0.822308192 1.6596029 1.52941294 0.80577843
## 193 -1.501833434 1.274989030 -0.5224844 -0.06074124 -1.50374851
## 136 1.052443947 -0.799892951 -0.6373311 0.17066179 1.26768382
## 100 -0.215813389 -0.142395045 -0.8670245 -0.78461740 -0.21041342
## 301 -0.752247165 1.915153117 -0.5224844 -0.29807769 -1.68851066
## 214 -1.288680940 0.071404563 -0.6373311 -0.77868399 0.06672981
## 363 0.981393116 -0.803264735 1.6596029 1.52941294 0.80577843
## 377 0.878369411 -1.163570739 1.6596029 1.52941294 0.80577843
## 480 0.690084708 -0.875639370 1.6596029 1.52941294 0.80577843
## 68 -1.675907970 1.283632195 -0.6373311 -0.37521203 0.20530143
## 31 0.906789743 0.207985566 -0.6373311 -0.60068166 1.17530274
## 160 1.116389695 -0.963638187 -0.5224844 -0.03107419 -1.73470120
## 28 0.718505041 0.312653344 -0.6373311 -0.60068166 1.17530274
## 117 0.139440767 -0.505740403 -0.4076377 0.14099473 -0.30279450
## 178 0.171413641 -0.226784628 -0.5224844 -0.66594918 -0.85708096
## 78 -0.809087830 0.140312434 -0.5224844 -0.06074124 0.11292035
## 34 0.938762618 -0.003724486 -0.6373311 -0.60068166 1.17530274
## 360 0.452064424 -0.610693120 1.6596029 1.52941294 0.80577843
## 191 -1.672355429 1.274989030 -0.5224844 -0.06074124 -1.50374851
## 482 0.224701764 -0.220041060 1.6596029 1.52941294 0.80577843
## 441 0.846396537 -0.916005799 1.6596029 1.52941294 0.80577843
## 127 0.960077867 -0.967769809 -0.8670245 -1.30675758 0.29768250
## 320 -0.347257427 0.098236367 -0.6373311 -0.61848189 -0.02565127
## 27 0.771793164 0.421215295 -0.6373311 -0.60068166 1.17530274
## 251 -1.974321461 1.710424083 -0.2927910 -0.46421320 0.29768250
## 39 -1.363284313 0.024816954 -0.5224844 -0.76681717 0.34387304
## 205 -1.302891107 0.628271335 -0.6373311 -1.09315478 -1.73470120
## 256 -1.757616426 2.576450217 -0.9818712 -0.55321437 -0.94946204
## 135 1.059549031 -0.688149176 -0.6373311 0.17066179 1.26768382
## 64 -0.894348827 1.987860180 -0.1779443 -0.73715011 0.57482574
## 451 0.853501620 -0.698786917 1.6596029 1.52941294 0.80577843
## 442 1.016918532 -0.821548354 1.6596029 1.52941294 0.80577843
## 355 -1.658145262 3.224877549 -0.6373311 -0.44047956 1.63720813
## 505 0.736267749 -0.667776001 -0.9818712 -0.80241764 1.17530274
## 40 -1.661697804 0.762715291 -0.7521778 -0.92701927 -0.07184181
## 359 0.526667797 -0.509254657 1.6596029 1.52941294 0.80577843
## 124 1.009813449 -0.878868684 -0.8670245 -1.30675758 0.29768250
## 164 0.899684660 -0.775530624 -0.5224844 -0.03107419 -1.73470120
## 30 0.665216917 0.210834961 -0.6373311 -0.60068166 1.17530274
## 497 0.153650933 -0.473209809 -0.4076377 -0.10227512 0.34387304
## 132 1.034681240 -0.723766614 -0.6373311 0.17066179 1.26768382
## 323 -0.663433626 0.439783853 -0.5224844 -0.71934988 0.52863520
## 85 -0.730931915 0.467470475 -0.7521778 -0.95668632 0.02053927
## 119 0.160756016 -0.625699934 -0.4076377 0.14099473 -0.30279450
## 447 0.988498199 -0.818271549 1.6596029 1.52941294 0.80577843
## 109 1.013365990 -0.646880437 -0.5224844 -0.14380900 1.12911220
## 432 0.913894827 -0.810578183 1.6596029 1.52941294 0.80577843
## 438 1.116389695 -0.893210639 1.6596029 1.52941294 0.80577843
## 313 0.775345706 -0.456398379 -0.6373311 -0.61848189 -0.02565127
## 174 0.551535588 -0.545536954 -0.5224844 -0.66594918 -0.85708096
## 74 -2.215894287 0.708671765 -0.6373311 -0.61254848 0.34387304
## 77 0.210491598 0.122123795 -0.5224844 -0.06074124 0.11292035
## 369 1.116389695 -1.169459489 1.6596029 1.52941294 0.80577843
## 115 0.555088129 -0.730652652 -0.4076377 0.14099473 -0.30279450
## 60 -0.759352248 1.489738437 -0.1779443 -0.73715011 0.57482574
## 444 1.116389695 -0.862722112 1.6596029 1.52941294 0.80577843
## 168 0.377461051 -0.650204732 -0.5224844 -0.03107419 -1.73470120
## 177 -0.759352248 -0.114043564 -0.5224844 -0.66594918 -0.85708096
## 217 -0.446728591 -0.324328918 -0.5224844 -0.78461740 -0.94946204
## 384 1.116389695 -1.074194714 1.6596029 1.52941294 0.80577843
## 173 0.707847416 -0.569376892 -0.5224844 -0.66594918 -0.85708096
## 229 -1.832219799 -0.199430436 -0.1779443 -0.60068166 -0.48755665
## 24 1.116389695 0.142544460 -0.6373311 -0.60068166 1.17530274
## 299 -1.722091011 1.915153117 -0.5224844 -0.29807769 -1.68851066
## 122 0.551535588 -0.758719193 -0.8670245 -1.30675758 0.29768250
## 186 0.007996729 -0.244735817 -0.7521778 -1.27709053 -0.30279450
## 42 -2.333128159 0.914588047 -0.7521778 -1.03975408 -0.25660396
## 279 -1.295786023 0.163962412 -0.6373311 -0.91515245 -0.39517558
## 331 -1.292233482 0.987105151 -0.6373311 0.12912791 -0.71850935
## 290 -1.622619847 1.672669599 -0.4076377 -0.68374941 -0.85708096
## 155 0.974288033 -0.971474023 -0.5224844 -0.03107419 -1.73470120
## 190 -1.054213197 0.366459420 -0.5224844 -0.06074124 -1.50374851
## 29 0.917447368 0.313270713 -0.6373311 -0.60068166 1.17530274
## 287 -1.317101273 2.514190935 -0.9818712 -0.99228679 -0.11803234
## 103 0.597718628 -0.513053850 -0.5224844 -0.14380900 1.12911220
## 286 -1.302891107 1.667968097 -0.9818712 -0.64221553 -1.45755797
## 483 0.299305137 -0.182571515 1.6596029 1.52941294 0.80577843
## 407 1.116389695 -1.242783921 1.6596029 1.52941294 0.80577843
## 338 -0.318837095 0.864296224 -0.5224844 -1.09315478 0.80577843
## 51 -0.812640371 1.434032764 -0.6373311 -0.98041997 -0.76469989
## 393 1.009813449 -0.961691100 1.6596029 1.52941294 0.80577843
## 118 0.498247464 -0.497524647 -0.4076377 0.14099473 -0.30279450
## 342 -0.684748875 1.540030259 -0.9818712 -0.73715011 -1.36517689
## 50 -0.233576097 1.088781064 -0.7521778 -1.03975408 -0.25660396
## 248 0.377461051 2.023287659 -0.2927910 -0.46421320 0.29768250
## 314 0.505352547 -0.252761613 -0.6373311 -0.61848189 -0.02565127
## 391 1.009813449 -0.887369379 1.6596029 1.52941294 0.80577843
## 321 -0.578172628 0.353969572 -0.5224844 -0.71934988 0.52863520
## 69 -1.128816570 1.283632195 -0.6373311 -0.37521203 0.20530143
## 399 1.116389695 -1.094852828 1.6596029 1.52941294 0.80577843
## 37 -0.254891346 -0.198100718 -0.5224844 -0.76681717 0.34387304
## 500 0.174966182 -0.662552110 -0.4076377 -0.10227512 0.34387304
## 258 0.651006751 -0.946969226 -0.5224844 -0.85581834 -2.51994036
## 324 0.203386515 0.439783853 -0.5224844 -0.71934988 0.52863520
## 242 -0.123447309 1.206746019 -0.4076377 -0.64221553 -0.85708096
## 148 0.963630408 -1.108529925 -0.5224844 -0.03107419 -1.73470120
## 83 -1.292233482 0.762525331 -0.6373311 -0.75495035 0.25149196
## 253 -2.194579038 2.427521836 -0.2927910 -0.46421320 0.29768250
## 156 0.498247464 -0.973326130 -0.5224844 -0.03107419 -1.73470120
## 387 1.116389695 -1.105490570 1.6596029 1.52941294 0.80577843
## 274 -0.595935336 0.271384606 -0.7521778 -1.09908819 0.06672981
## 67 -1.331311439 1.337533251 -0.6373311 -0.42267932 -1.08803366
## 366 0.686532167 -1.036155290 1.6596029 1.52941294 0.80577843
## 114 0.952972784 -0.592219542 -0.4076377 0.14099473 -0.30279450
## 162 0.789555872 -0.866283856 -0.5224844 -0.03107419 -1.73470120
## 312 -0.560409921 -0.548386349 -0.6373311 -0.61848189 -0.02565127
## 73 -2.159053622 0.708671765 -0.6373311 -0.61254848 0.34387304
## 406 1.116389695 -1.125341355 1.6596029 1.52941294 0.80577843
## 322 -0.507121797 0.353969572 -0.5224844 -0.71934988 0.52863520
## 439 0.686532167 -0.937661202 1.6596029 1.52941294 0.80577843
## 167 0.981393116 -0.830666418 -0.5224844 -0.03107419 -1.73470120
## 218 0.587061003 -0.177585074 -0.5224844 -0.78461740 -0.94946204
## 308 0.061284852 -0.290801037 -0.2927910 -1.10502160 -0.02565127
## 472 0.786003330 -0.330407628 1.6596029 1.52941294 0.80577843
## 365 0.508905089 -0.897722181 1.6596029 1.52941294 0.80577843
## 120 -0.119894767 -0.491920837 -0.4076377 0.14099473 -0.30279450
## 488 -0.546199754 -0.305237972 1.6596029 1.52941294 0.80577843
## 112 0.462722049 -0.530720099 -0.4076377 0.14099473 -0.30279450
## 113 0.864159245 -0.684634922 -0.4076377 0.14099473 -0.30279450
## 354 -1.153684361 3.956602197 -0.5224844 -1.31269099 -0.67231881
## 334 -1.082633530 1.264826188 -0.5224844 -1.09315478 0.80577843
## 11 0.913894827 1.211779950 -0.5224844 -0.57694801 -1.50374851
## 348 -1.452097852 2.251144283 -0.6373311 -0.33961157 -0.25660396
## 340 -0.823297996 0.483047168 -0.5224844 -1.09315478 0.80577843
## 2 0.366803426 0.556609050 -0.8670245 -0.98635338 -0.30279450
## 57 -1.167894527 2.560921014 -0.8670245 -0.56508119 -0.53374719
## 268 -0.055949019 -0.652246798 -0.5224844 -0.85581834 -2.51994036
## 449 1.070206655 -0.728230666 1.6596029 1.52941294 0.80577843
## 230 -1.675907970 -0.199430436 -0.1779443 -0.60068166 -0.48755665
## 332 -1.427230061 1.351400307 -0.9818712 -0.61848189 -0.71850935
## 7 -0.070159185 0.838414219 -0.5224844 -0.57694801 -1.50374851
## 170 0.945867701 -0.727803257 -0.5224844 -0.03107419 -1.73470120
## 198 -1.135921653 1.668775425 -0.8670245 -0.47014661 -2.70470251
## 203 -1.878402839 1.175355183 -0.8670245 -0.35741180 -1.73470120
## 201 -1.942348587 1.832330701 -0.7521778 -0.03700760 -0.67231881
## 188 -0.976057283 -0.003059627 -0.5224844 -0.06074124 -1.50374851
## 149 0.896132119 -1.075856861 -0.5224844 -0.03107419 -1.73470120
## 204 -1.256708066 0.628271335 -0.6373311 -1.09315478 -1.73470120
## 309 0.494694923 -0.226784628 -0.6373311 -0.61848189 -0.02565127
## 22 0.732715207 0.103175318 -0.6373311 -0.60068166 1.17530274
## 82 0.064837394 0.762525331 -0.6373311 -0.75495035 0.25149196
## 97 0.036417061 -0.142395045 -0.8670245 -0.78461740 -0.21041342
## 259 1.116389695 -0.902518663 -0.5224844 -0.85581834 -2.51994036
## 468 0.920999910 -0.595876266 1.6596029 1.52941294 0.80577843
## 138 1.059549031 -0.923794146 -0.6373311 0.17066179 1.26768382
## 481 -0.137657475 -0.176112886 1.6596029 1.52941294 0.80577843
## 329 -1.519596142 0.674146595 -0.6373311 0.12912791 -0.71850935
## 106 0.999155824 -0.801697568 -0.5224844 -0.14380900 1.12911220
## 63 -0.027528687 1.629121344 -0.1779443 -0.73715011 0.57482574
## 199 -1.075528447 1.668775425 -0.8670245 -0.47014661 -2.70470251
## 479 0.999155824 -0.771493981 1.6596029 1.52941294 0.80577843
## 125 0.967182950 -0.849472425 -0.8670245 -1.30675758 0.29768250
## 130 0.928104993 -0.862009763 -0.6373311 0.17066179 1.26768382
## 416 1.116389695 -0.930965123 1.6596029 1.52941294 0.80577843
## 380 1.116389695 -1.144004893 1.6596029 1.52941294 0.80577843
## 108 0.590613545 -0.794336631 -0.5224844 -0.14380900 1.12911220
## 171 0.924552451 -0.650204732 -0.5224844 -0.03107419 -1.73470120
## 437 0.878369411 -0.851229552 1.6596029 1.52941294 0.80577843
## 252 -2.119975665 1.710424083 -0.2927910 -0.46421320 0.29768250
## 491 1.045338864 -0.936711403 -0.6373311 1.79641644 0.75958789
## 91 -0.087921893 -0.333731922 -0.8670245 -0.82021787 -0.30279450
## 75 -2.222999370 0.216771201 -0.5224844 -0.06074124 0.11292035
## 412 1.116389695 -1.076854150 1.6596029 1.52941294 0.80577843
## 351 -0.858823412 2.373098390 -0.9818712 -0.43454615 0.57482574
## 18 0.466274590 0.219810555 -0.6373311 -0.60068166 1.17530274
## 89 0.629691502 -0.177300134 -0.8670245 -0.82021787 -0.30279450
## 283 -0.670538709 0.672864367 -0.5224844 -1.14062207 -1.64232012
## 269 -0.567515004 -0.438352210 -0.5224844 -0.85581834 -2.51994036
## 196 -1.299338565 0.880157857 -0.6373311 -0.90921904 -1.87327282
## 364 0.725610124 -0.897722181 1.6596029 1.52941294 0.80577843
## 263 0.814423663 -0.715455878 -0.5224844 -0.85581834 -2.51994036
## 126 0.704294875 -0.855836074 -0.8670245 -1.30675758 0.29768250
## 163 1.052443947 -0.833135893 -0.5224844 -0.03107419 -1.73470120
## 131 1.077311738 -0.796188738 -0.6373311 0.17066179 1.26768382
## 465 -0.112789684 -0.394946426 1.6596029 1.52941294 0.80577843
## 422 0.949420242 -0.912016646 1.6596029 1.52941294 0.80577843
## 495 -0.922769160 -0.671195275 -0.4076377 -0.10227512 0.34387304
## 207 -0.571067545 0.265875775 -0.6373311 -0.77868399 0.06672981
## 285 -1.697223220 1.667968097 -0.9818712 -0.73121670 -1.45755797
## 325 -1.011582699 0.439783853 -0.5224844 -0.71934988 0.52863520
## 151 1.020471073 -1.033875774 -0.5224844 -0.03107419 -1.73470120
## 169 0.977840575 -0.804974372 -0.5224844 -0.03107419 -1.73470120
## 316 0.324172928 0.071214604 -0.6373311 -0.61848189 -0.02565127
## 401 1.116389695 -1.047742830 1.6596029 1.52941294 0.80577843
## 76 -0.837508162 0.336018383 -0.5224844 -0.06074124 0.11292035
## 247 -1.196314860 2.023287659 -0.2927910 -0.46421320 0.29768250
## 423 0.675874542 -0.875639370 1.6596029 1.52941294 0.80577843
## 368 1.116389695 -1.084879946 1.6596029 1.52941294 0.80577843
## 296 -1.331311439 1.028326399 -0.6373311 -0.70748306 -1.13422420
## 139 1.052443947 -1.009845876 -0.6373311 0.17066179 1.26768382
## 371 1.027576156 -1.231243871 1.6596029 1.52941294 0.80577843
## 370 1.002708366 -1.157966929 1.6596029 1.52941294 0.80577843
## 235 -0.073711727 -0.067978344 -0.1779443 -0.60068166 -0.48755665
## 238 0.107467893 0.167619136 -0.1779443 -0.60068166 -0.48755665
## 134 0.952972784 -0.629309168 -0.6373311 0.17066179 1.26768382
## 298 -0.375677759 1.199100142 -0.6373311 -0.70748306 -1.13422420
## 440 0.899684660 -0.939275859 1.6596029 1.52941294 0.80577843
## 306 -0.372125218 -0.201852421 -0.2927910 -1.10502160 -0.02565127
## 197 -1.224735192 1.668775425 -0.8670245 -0.47014661 -2.70470251
## 152 1.116389695 -1.046413113 -0.5224844 -0.03107419 -1.73470120
## 390 1.077311738 -0.981589375 1.6596029 1.52941294 0.80577843
## 224 0.434301716 -0.248345051 -0.1779443 -0.60068166 -0.48755665
## 385 0.803766038 -1.118645277 1.6596029 1.52941294 0.80577843
## 17 -1.395257187 0.334118787 -0.6373311 -0.60068166 1.17530274
## 105 0.761135540 -0.652531738 -0.5224844 -0.14380900 1.12911220
## 358 0.796660955 -0.612545227 1.6596029 1.52941294 0.80577843
## 154 1.063101572 -1.031406299 -0.5224844 -0.03107419 -1.73470120
## 58 -0.997372532 2.151178006 -0.5224844 -0.90328562 -1.54993904
## 129 1.073759197 -0.941507885 -0.6373311 0.17066179 1.26768382
## 388 0.743372832 -1.081175732 1.6596029 1.52941294 0.80577843
## 208 0.146545850 0.265875775 -0.6373311 -0.77868399 0.06672981
## 471 0.547983046 -0.361703483 1.6596029 1.52941294 0.80577843
## 221 0.707847416 -0.443243672 -0.1779443 -0.60068166 -0.48755665
## 49 0.949420242 0.985395514 -0.7521778 -1.03975408 -0.25660396
## 144 1.116389695 -1.131799984 -0.5224844 -0.03107419 -1.73470120
## 157 0.903237202 -0.977647712 -0.5224844 -0.03107419 -1.73470120
## 101 0.402328842 -0.483087712 -0.5224844 -0.14380900 1.12911220
## 462 0.704294875 -0.583148968 1.6596029 1.52941294 0.80577843
## 227 0.636796585 -0.275129364 -0.1779443 -0.60068166 -0.48755665
## 222 0.807318580 -0.354769955 -0.1779443 -0.60068166 -0.48755665
## 219 0.896132119 -0.430136454 -0.5224844 -0.78461740 -0.94946204
## 494 -0.517779422 -0.671195275 -0.4076377 -0.10227512 0.34387304
## 159 1.116389695 -0.967722319 -0.5224844 -0.03107419 -1.73470120
## 140 1.041786323 -1.009798386 -0.6373311 0.17066179 1.26768382
## 59 -1.398809729 1.908979428 -0.1779443 -0.73715011 0.57482574
## 463 0.512457630 -0.503698336 1.6596029 1.52941294 0.80577843
## 71 -2.201684121 0.708671765 -0.6373311 -0.61254848 0.34387304
## 278 -1.455650394 0.507077066 -0.6373311 -0.91515245 -0.39517558
## 339 -1.111053862 0.483047168 -0.5224844 -1.09315478 0.80577843
## 375 1.116389695 -1.262302277 1.6596029 1.52941294 0.80577843
## 397 0.974288033 -1.005951703 1.6596029 1.52941294 0.80577843
## 94 -1.409467353 -0.061329756 -0.6373311 -0.82021787 -0.11803234
## 467 0.572850837 -0.438589660 1.6596029 1.52941294 0.80577843
## 257 -1.221182651 1.206746019 -0.7521778 -0.97448656 -1.18041474
## 378 1.073759197 -1.157349560 1.6596029 1.52941294 0.80577843
## 38 -0.961847117 0.066085693 -0.5224844 -0.76681717 0.34387304
## 65 -0.322389636 2.577684955 -0.7521778 -1.14062207 0.06672981
## 295 -0.933426784 0.810965047 -0.6373311 -0.70748306 -1.13422420
## 26 0.608376252 0.313223223 -0.6373311 -0.60068166 1.17530274
## 415 1.116389695 -1.014784828 1.6596029 1.52941294 0.80577843
## 146 1.116389695 -1.128333220 -0.5224844 -0.03107419 -1.73470120
## 476 1.024023615 -0.754635060 1.6596029 1.52941294 0.80577843
## 147 1.116389695 -1.082030551 -0.5224844 -0.03107419 -1.73470120
## 327 -1.409467353 0.769743799 -0.5224844 -0.71934988 0.52863520
## 346 -0.713169208 2.003389383 -0.7521778 -0.33367816 0.15911089
## 424 0.587061003 -0.842111488 1.6596029 1.52941294 0.80577843
## 271 -0.940531867 0.301065804 -0.7521778 -1.09908819 0.06672981
## 474 -0.034633770 -0.599390520 1.6596029 1.52941294 0.80577843
## 466 -0.723826832 -0.345984321 1.6596029 1.52941294 0.80577843
## 289 -0.816192913 1.672669599 -0.4076377 -0.68374941 -0.85708096
## 47 -1.253155525 0.619913109 -0.7521778 -1.03975408 -0.25660396
## 111 -0.503569256 -0.483087712 -0.5224844 -0.14380900 1.12911220
## 445 0.995603282 -0.902043764 1.6596029 1.52941294 0.80577843
## 184 0.960077867 -0.450224689 -0.7521778 -1.27709053 -0.30279450
## 176 -1.260260608 -0.314735955 -0.5224844 -0.66594918 -0.85708096
## 434 0.686532167 -0.702491131 1.6596029 1.52941294 0.80577843
## 35 1.006260907 -0.016736723 -0.6373311 -0.60068166 1.17530274
## 141 0.889027036 -1.036772659 -0.6373311 0.17066179 1.26768382
## 456 0.636796585 -0.645503230 1.6596029 1.52941294 0.80577843
## 215 -2.088002791 -0.098561851 -0.6373311 -0.77868399 0.06672981
## 96 -0.382782843 -0.142395045 -0.8670245 -0.78461740 -0.21041342
## 464 0.757582998 -0.471785112 1.6596029 1.52941294 0.80577843
## 226 0.512457630 -0.427714469 -0.1779443 -0.60068166 -0.48755665
## 116 0.697189791 -0.632538482 -0.4076377 0.14099473 -0.30279450
## 137 0.885474494 -0.868183453 -0.6373311 0.17066179 1.26768382
## 202 -1.071975905 1.175355183 -0.8670245 -0.35741180 -1.73470120
## 293 -1.604857139 0.627653966 -0.6373311 -0.96855315 0.34387304
## 457 0.686532167 -0.576737829 1.6596029 1.52941294 0.80577843
## 12 0.508905089 1.154792049 -0.5224844 -0.57694801 -1.50374851
## 304 -1.807352008 0.805741156 -0.2927910 -0.47014661 -1.08803366
## 357 1.024023615 -0.794431611 1.6596029 1.52941294 0.80577843
## 446 0.931657534 -0.858210570 1.6596029 1.52941294 0.80577843
## 41 -1.874850298 0.762715291 -0.7521778 -0.92701927 -0.07184181
## 237 0.281542429 0.167619136 -0.1779443 -0.60068166 -0.48755665
## 335 -1.068423364 1.264826188 -0.5224844 -1.09315478 0.80577843
## 383 1.116389695 -1.051731983 1.6596029 1.52941294 0.80577843
## 273 -0.350809969 0.058154876 -0.7521778 -1.09908819 0.06672981
## 315 0.665216917 -0.091533343 -0.6373311 -0.61848189 -0.02565127
## 272 -1.857087590 0.301065804 -0.7521778 -1.09908819 0.06672981
## 461 0.761135540 -0.568712033 1.6596029 1.52941294 0.80577843
## 395 0.928104993 -0.955944820 1.6596029 1.52941294 0.80577843
## 414 1.116389695 -1.047457891 1.6596029 1.52941294 0.80577843
## 104 0.668769459 -0.513053850 -0.5224844 -0.14380900 1.12911220
## 356 -1.743406260 3.224877549 -0.6373311 -0.44047956 1.63720813
## 254 -2.137738373 2.427521836 -0.2927910 -0.46421320 0.29768250
## 275 -1.267365691 0.134186234 -0.6373311 -0.91515245 -0.39517558
## 102 0.096810268 -0.445903107 -0.5224844 -0.14380900 1.12911220
## 484 -1.004477616 0.144016647 1.6596029 1.52941294 0.80577843
## 475 0.952972784 -0.648352625 1.6596029 1.52941294 0.80577843
## 133 1.041786323 -0.696982300 -0.6373311 0.17066179 1.26768382
## 350 -1.210525026 2.373098390 -0.9818712 -0.43454615 0.57482574
## 45 -1.015135240 0.914588047 -0.7521778 -1.03975408 -0.25660396
## 62 0.881921953 1.435837380 -0.1779443 -0.73715011 0.57482574
## 496 -1.413019895 -0.473209809 -0.4076377 -0.10227512 0.34387304
## 32 1.116389695 0.180441414 -0.6373311 -0.60068166 1.17530274
## 398 1.077311738 -1.026562327 1.6596029 1.52941294 0.80577843
## 46 -1.235392817 0.619913109 -0.7521778 -1.03975408 -0.25660396
## 372 1.116389695 -1.247058014 1.6596029 1.52941294 0.80577843
## 234 0.064837394 -0.067978344 -0.1779443 -0.60068166 -0.48755665
## 400 0.327725469 -1.089723917 1.6596029 1.52941294 0.80577843
## 80 -1.135921653 0.336018383 -0.5224844 -0.06074124 0.11292035
## 493 0.530220338 -0.800272871 -0.6373311 1.79641644 0.75958789
## 418 0.729162665 -1.019866249 1.6596029 1.52941294 0.80577843
## 98 0.263779721 -0.142395045 -0.8670245 -0.78461740 -0.21041342
## 337 -0.791325122 0.681982432 -0.5224844 -1.09315478 0.80577843
## 443 1.116389695 -0.850184774 1.6596029 1.52941294 0.80577843
## 245 0.281542429 1.975512802 -0.2927910 -0.46421320 0.29768250
## 88 -0.418308258 -0.022530493 -0.7521778 -0.95668632 0.02053927
## 302 -1.000925074 0.805741156 -0.2927910 -0.47014661 -1.08803366
## 70 -1.263813149 1.283632195 -0.6373311 -0.37521203 0.20530143
## 429 0.338383094 -0.883047797 1.6596029 1.52941294 0.80577843
## 489 0.857054162 -0.937518732 -0.6373311 1.79641644 0.75958789
## 121 0.039969603 -0.730082773 -0.8670245 -1.30675758 0.29768250
## 14 -0.240681180 0.433325224 -0.6373311 -0.60068166 1.17530274
## 209 -0.336599802 0.210929941 -0.6373311 -0.77868399 0.06672981
## 161 0.853501620 -0.948203963 -0.5224844 -0.03107419 -1.73470120
## 431 0.622586419 -0.827437103 1.6596029 1.52941294 0.80577843
## 9 1.116389695 1.086121629 -0.5224844 -0.57694801 -1.50374851
## 473 0.228254306 -0.426717180 1.6596029 1.52941294 0.80577843
## 179 0.206939056 -0.417789076 -0.5224844 -0.66594918 -0.85708096
## 4 -0.809087830 1.076671135 -0.7521778 -1.10502160 0.11292035
## 54 -1.675907970 1.434032764 -0.6373311 -0.98041997 -0.76469989
## 318 0.111020434 0.112388362 -0.6373311 -0.61848189 -0.02565127
## 386 1.048891406 -1.125008926 1.6596029 1.52941294 0.80577843
## 16 -0.428965883 0.334118787 -0.6373311 -0.60068166 1.17530274
## 404 0.974288033 -0.993604324 1.6596029 1.52941294 0.80577843
## 281 -0.144762558 0.427246515 -0.5224844 -1.14062207 -1.64232012
## 79 -0.528437047 0.578929311 -0.5224844 -0.06074124 0.11292035
## 175 0.004444187 -0.519132560 -0.5224844 -0.66594918 -0.85708096
## 250 -1.814457091 1.914535748 -0.2927910 -0.46421320 0.29768250
## 185 0.754030456 -0.383311396 -0.7521778 -1.27709053 -0.30279450
## 15 0.565745754 0.316689987 -0.6373311 -0.60068166 1.17530274
## 408 1.116389695 -1.191922219 1.6596029 1.52941294 0.80577843
## 3 -0.265548971 0.556609050 -0.8670245 -0.98635338 -0.30279450
## 345 -1.437887686 1.268150482 -0.5224844 -0.22687676 -0.39517558
## 93 -0.531989588 -0.061329756 -0.6373311 -0.82021787 -0.11803234
## 326 -1.913928255 0.769743799 -0.5224844 -0.71934988 0.52863520
## 55 -0.745142082 1.673856847 -0.7521778 0.36053095 1.22149328
## 233 0.167861099 0.020590352 -0.1779443 -0.60068166 -0.48755665
## 23 0.821528746 0.086363887 -0.6373311 -0.60068166 1.17530274
## 183 0.839291454 -0.519749928 -0.7521778 -1.27709053 -0.30279450
## 307 0.118125517 -0.330455118 -0.2927910 -1.10502160 -0.02565127
## 336 -1.210525026 1.040151389 -0.5224844 -1.09315478 0.80577843
## black lstat medv crime
## 223 0.369308489 -0.381324714 0.540082359 med_high
## 419 -3.726650277 1.115651608 -1.493166348 high
## 433 -2.833938506 -0.087250788 -0.699438243 high
## 240 0.296905730 -0.739814835 0.083416874 med_low
## 459 -0.925178346 0.500897066 -0.829914096 high
## 303 0.295043632 -0.557769070 0.420479494 med_low
## 61 0.421009097 0.069588640 -0.416740562 med_low
## 212 0.422433054 1.586169891 -0.351502635 med_high
## 352 0.154509988 -1.003081017 0.170400776 low
## 376 0.440615895 0.110198849 -0.819041108 high
## 280 0.440615895 -1.092703547 1.366429428 med_low
## 56 0.429990982 -1.098304955 1.399048391 low
## 189 0.286609423 -1.133313756 0.790161078 med_low
## 262 0.347182381 -0.755218707 2.236268447 med_high
## 294 0.440615895 -0.570372239 0.148654801 med_low
## 485 0.153962312 0.096195329 -0.210153795 med_high
## 241 0.378728515 -0.178273670 -0.057931966 med_low
## 210 0.440615895 1.461538560 -0.275391721 med_high
## 25 0.412465352 0.510699530 -0.753803182 med_high
## 492 0.366241503 0.758561840 -0.971262937 med_low
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## 356 0.212125496 -0.991878200 -0.210153795 med_low
## 254 0.440615895 -1.276149663 2.203649484 med_high
## 275 0.440615895 -1.277550015 1.072858759 low
## 102 0.426157250 -0.697804274 0.431352482 med_low
## 484 0.397020891 -0.312707465 -0.079677942 med_high
## 475 -0.044844052 0.768364304 -0.949516961 high
## 133 0.318593697 -0.214682823 0.050797911 med_high
## 350 0.363393588 -0.947066935 0.442225470 low
## 45 0.358354970 -0.434538092 -0.144915868 med_low
## 62 0.234470674 0.250234052 -0.710311231 med_low
## 496 0.401073693 0.692745294 0.061670899 med_low
## 32 0.219683424 0.054184768 -0.873406047 med_high
## 398 0.398992524 1.017626966 -1.525785311 high
## 46 0.440615895 -0.342114857 -0.351502635 med_low
## 372 0.103795196 -0.437338796 2.986504601 high
## 234 0.244000235 -1.218735230 2.801663810 med_high
## 400 -0.202793791 2.424980758 -1.764991042 high
## 80 0.431414939 -0.497553933 -0.242772758 med_low
## 493 0.440615895 0.097595681 -0.264518733 med_low
## 418 -2.511795523 1.958663532 -1.319198544 high
## 98 0.440615895 -1.182326077 1.757856987 med_low
## 337 0.440615895 -0.399529291 -0.329756660 low
## 443 0.427362137 0.551309739 -0.449359525 high
## 245 0.173240505 -0.021434242 -0.536343427 med_low
## 88 0.421447237 -0.589977167 -0.036185991 low
## 302 0.428019349 -0.441539852 -0.057931966 low
## 70 0.440615895 -0.540964846 -0.177534831 med_low
## 429 -2.847301799 1.241683291 -1.253960618 high
## 489 0.420790026 0.757161488 -0.797295133 med_low
## 121 0.355726125 0.240431588 -0.057931966 low
## 14 0.440615895 -0.615183504 -0.231899770 med_high
## 209 0.269960074 0.281041797 0.203019739 med_low
## 161 -0.194469117 -1.001680665 0.485717421 med_high
## 431 -2.992764527 0.698346703 -0.873406047 high
## 9 0.328123258 2.419379350 -0.655946292 med_low
## 473 0.401949974 0.239031236 0.072543887 med_high
## 179 0.378947585 -0.802830676 0.801034065 low
## 4 0.415751408 -1.360170785 1.181588636 low
## 54 0.440615895 -0.591377519 0.094289862 low
## 318 0.440615895 0.460286857 -0.297137697 med_low
## 386 0.440615895 2.542610329 -1.667134152 high
## 16 0.426595391 -0.585776111 -0.286264709 med_high
## 404 0.440615895 0.996621685 -1.547531287 high
## 281 0.335571651 -1.245341918 2.486347165 low
## 79 0.325603949 -0.043839875 -0.144915868 low
## 175 0.400416482 -0.421934923 0.007305960 med_low
## 250 0.406002776 -0.853243349 0.398733519 med_low
## 185 0.375990135 0.185817859 0.420479494 med_low
## 15 0.255720500 -0.335113097 -0.471105500 med_high
## 408 -0.269281649 -0.073247267 0.583574310 high
## 3 0.396035074 -1.207532413 1.322937477 low
## 345 0.342800973 -1.126311996 0.942382906 low
## 93 0.419913745 -0.629187024 0.039924923 low
## 326 0.405345565 -1.060495450 0.224765715 med_low
## 55 0.440615895 0.300646725 -0.394994586 low
## 233 0.320236725 -1.425987330 2.084046619 med_high
## 23 0.440615895 0.849584722 -0.797295133 med_high
## 183 0.410165113 -1.096904603 1.670873085 med_low
## 307 0.440615895 -0.865846518 1.181588636 low
## 336 0.440615895 -0.650192305 -0.155788856 low
## zn indus chas nox rm age
## 1 0.28454827 -1.28663623 -0.2723291 -0.14407485 0.41326292 -0.11989477
## 5 -0.48724019 -1.30558569 -0.2723291 -0.83445805 1.22736204 -0.51067434
## 6 -0.48724019 -1.30558569 -0.2723291 -0.83445805 0.20689164 -0.35080997
## 8 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.16030692 0.97784057
## 10 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.39941295 0.61548134
## 19 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.17935407 -1.13592165
## 20 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.79365326 0.03286452
## 21 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.01710355 1.04889141
## 33 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.47626846 0.47693221
## 43 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.16457667 -2.20168412
## 44 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.10480016 -2.20523666
## 48 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.36240845 0.60127117
## 52 0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.24143218 -0.19805068
## 72 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.46061271 -1.81445709
## 84 0.58468822 -0.91493524 -0.2723291 -1.11061133 -0.16742317 -0.77711496
## 86 -0.48724019 -0.96886832 -0.2723291 -0.91212616 0.49154168 -0.44317605
## 90 -0.48724019 -1.12629463 -0.2723291 -0.56693456 1.13058103 -0.19449814
## 92 -0.48724019 -1.12629463 -0.2723291 -0.56693456 0.17131038 0.18917635
## 99 -0.48724019 -1.20209248 -0.2723291 -0.94664532 2.18520944 -1.12526403
## 107 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.63851899 0.82863383
## 110 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.07918165 0.80376604
## 123 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.46061271 0.86415924
## 142 -0.48724019 1.56744433 -0.2723291 0.59808708 -1.80131441 1.11638970
## 143 -0.48724019 1.23072696 3.6647712 2.72964520 -1.25478633 1.11638970
## 145 -0.48724019 1.23072696 -0.2723291 2.72964520 -1.96641144 1.03823378
## 150 -0.48724019 1.23072696 -0.2723291 2.72964520 -0.97867579 0.93521008
## 158 -0.48724019 1.23072696 -0.2723291 0.43412107 0.93701900 1.02402361
## 166 -0.48724019 1.23072696 -0.2723291 0.43412107 -0.26135768 0.86771179
## 180 -0.48724019 -1.26477147 -0.2723291 -0.57556435 0.98967926 -0.36146759
## 181 -0.48724019 -1.26477147 -0.2723291 -0.57556435 2.10693068 0.52311526
## 187 -0.48724019 -1.26477147 -0.2723291 -0.57556435 2.20086519 -0.53198959
## 192 1.44223095 -1.12192167 -0.2723291 -1.01568364 0.64667596 -1.34196906
## 194 2.08538800 -1.19626187 -0.2723291 -1.32635608 0.73349422 -2.08445025
## 195 2.08538800 -1.19626187 -0.2723291 -1.32635608 0.45453718 -1.76827405
## 200 3.58608778 -1.40907891 -0.2723291 -1.30909650 0.98256301 -1.89261301
## 213 -0.48724019 -0.07970124 3.6647712 -0.56693456 -0.67979324 -0.52488450
## 225 -0.48724019 -0.71961001 -0.2723291 -0.43748771 2.81997903 0.34548818
## 231 -0.48724019 -0.71961001 -0.2723291 -0.43748771 -0.43214771 -0.01687106
## 232 -0.48724019 -0.71961001 -0.2723291 -0.43748771 1.60452335 0.29575260
## 236 -0.48724019 -0.71961001 -0.2723291 -0.41159834 -0.28270643 -0.25133880
## 239 0.79907391 -0.90473168 -0.2723291 -1.09335175 0.27947740 -1.77893168
## 243 0.79907391 -0.90473168 -0.2723291 -1.09335175 0.10441762 -0.55685738
## 246 0.45605682 -0.76917014 -0.2723291 -1.06746238 -0.96728979 0.05773231
## 249 0.45605682 -0.76917014 -0.2723291 -1.06746238 0.21116139 -0.69185396
## 260 0.37030254 -1.04466617 -0.2723291 0.79657225 0.79327073 1.11638970
## 261 0.37030254 -1.04466617 -0.2723291 0.79657225 1.30706406 0.46982713
## 265 0.37030254 -1.04466617 -0.2723291 0.79657225 1.31133381 0.81797620
## 267 0.37030254 -1.04466617 -0.2723291 0.79657225 1.03806976 0.56929830
## 282 0.37030254 -1.13795583 -0.2723291 -0.96476788 0.97260025 -1.11460640
## 284 3.37170210 -1.44697784 3.6647712 -1.32635608 2.33180421 -1.55512156
## 291 2.94293073 -0.90181638 -0.2723291 -1.24005818 0.82031248 -1.44499277
## 297 -0.48724019 0.40569652 -0.2723291 -1.01568364 0.37625841 -0.62435567
## 305 0.92770532 -1.30558569 -0.2723291 -0.71364099 1.35403131 -0.97605728
## 310 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -0.44495696 0.28864751
## 311 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -1.86678392 -1.09329115
## 317 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -0.52750547 0.51956271
## 319 -0.48724019 -0.18027916 -0.2723291 -0.09229612 0.13857563 -0.04884394
## 330 -0.48724019 -1.15107469 -0.2723291 -0.81719847 0.06883637 -1.82511472
## 333 1.01345959 -0.74001712 -0.2723291 -1.00791683 -0.36098520 -1.60840968
## 341 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.45064996 -0.35791505
## 343 -0.48724019 -1.34785757 -0.2723291 -0.31667065 0.36344916 -0.31528455
## 344 1.87100232 -1.07236154 -0.2723291 -0.61008351 0.58547620 -0.43251842
## 347 -0.48724019 -0.98344483 -0.2723291 -0.97253469 -0.55027747 -0.57817263
## 362 -0.48724019 1.01499462 -0.2723291 1.85803641 -0.04787015 0.80021350
## 373 -0.48724019 1.01499462 3.6647712 0.97779784 -0.58301223 0.74692537
## 379 -0.48724019 1.01499462 -0.2723291 1.00368721 0.13572913 0.98139312
## 392 -0.48724019 1.01499462 -0.2723291 1.25395112 -0.33252019 0.49469492
## 394 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.13041866 0.85350162
## 402 -0.48724019 1.01499462 -0.2723291 1.19354259 0.08306887 1.11638970
## 403 -0.48724019 1.01499462 -0.2723291 1.19354259 0.16988713 1.11638970
## 405 -0.48724019 1.01499462 -0.2723291 1.19354259 -1.07261030 0.59771863
## 409 -0.48724019 1.01499462 -0.2723291 0.36508275 -0.95021078 1.04178632
## 411 -0.48724019 1.01499462 -0.2723291 0.36508275 -0.75095575 1.11638970
## 413 -0.48724019 1.01499462 -0.2723291 0.36508275 -2.35780525 1.11638970
## 417 -0.48724019 1.01499462 -0.2723291 1.07272553 0.70787571 0.78955587
## 420 -0.48724019 1.01499462 -0.2723291 1.40928733 0.76765222 0.28154243
## 421 -0.48724019 1.01499462 -0.2723291 1.40928733 0.17984989 1.11638970
## 427 -0.48724019 1.01499462 -0.2723291 0.25289548 -0.63709574 -0.31528455
## 428 -0.48724019 1.01499462 -0.2723291 1.07272553 -0.11760941 0.35969834
## 430 -0.48724019 1.01499462 -0.2723291 1.07272553 0.13572913 0.96007787
## 435 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.10906991 0.93876262
## 436 -0.48724019 1.01499462 -0.2723291 1.59914271 0.49011843 0.92455245
## 448 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.04787015 0.99560328
## 450 -0.48724019 1.01499462 -0.2723291 1.36613839 0.18838939 1.05599649
## 452 -0.48724019 1.01499462 -0.2723291 1.36613839 0.52712294 1.05244395
## 453 -0.48724019 1.01499462 -0.2723291 1.36613839 0.01759936 0.82508129
## 454 -0.48724019 1.01499462 -0.2723291 1.36613839 1.57748160 1.09152190
## 455 -0.48724019 1.01499462 -0.2723291 1.36613839 0.63102020 0.90678974
## 458 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.49619397 0.41653901
## 469 -0.48724019 1.01499462 -0.2723291 0.21837632 -0.51042647 0.08615264
## 477 -0.48724019 1.01499462 -0.2723291 0.51178918 0.28374715 0.88902704
## 478 -0.48724019 1.01499462 -0.2723291 0.51178918 -1.39568810 1.02047107
## 486 -0.48724019 1.01499462 -0.2723291 0.24426569 0.03894811 -0.59238279
## 487 -0.48724019 1.01499462 -0.2723291 0.24426569 -0.24285543 0.39877630
## 490 -0.48724019 2.42017014 -0.2723291 0.46864023 -1.23913058 1.05599649
## 498 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.69829550 0.07194248
## 499 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.37806420 -0.11634223
## 501 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.36667820 0.39522376
## 502 -0.48724019 0.11562398 -0.2723291 0.15796779 0.43888142 0.01865435
## 503 -0.48724019 0.11562398 -0.2723291 0.15796779 -0.23431593 0.28864751
## 504 -0.48724019 0.11562398 -0.2723291 0.15796779 0.98398626 0.79666096
## 506 -0.48724019 0.11562398 -0.2723291 0.15796779 -0.36240845 0.43430172
## dis rad tax ptratio black
## 1 0.1400749840 -0.9818712 -0.66594918 -1.45755797 0.44061589
## 5 1.0766711351 -0.7521778 -1.10502160 0.11292035 0.44061589
## 6 1.0766711351 -0.7521778 -1.10502160 0.11292035 0.41016511
## 8 1.0236248974 -0.5224844 -0.57694801 -1.50374851 0.44061589
## 10 1.3283202075 -0.5224844 -0.57694801 -1.50374851 0.32899954
## 19 0.0006920764 -0.6373311 -0.60068166 1.17530274 -0.74137830
## 20 0.0006920764 -0.6373311 -0.60068166 1.17530274 0.37544246
## 21 0.0013569352 -0.6373311 -0.60068166 1.17530274 0.21793086
## 33 0.0925850666 -0.6373311 -0.60068166 1.17530274 -1.35904722
## 43 0.9145880470 -0.7521778 -1.03975408 -0.25660396 0.29241479
## 44 0.9145880470 -0.7521778 -1.03975408 -0.25660396 0.41388931
## 48 0.8996287230 -0.7521778 -1.03975408 -0.25660396 0.39504926
## 52 1.4340327636 -0.6373311 -0.98041997 -0.76469989 0.40852209
## 72 0.7086717651 -0.6373311 -0.61254848 0.34387304 0.22198366
## 84 0.7625253315 -0.6373311 -0.75495035 0.25149196 0.37204687
## 86 0.3051974268 -0.7521778 -0.95668632 0.02053927 0.39022971
## 90 -0.1807194081 -0.8670245 -0.82021787 -0.30279450 0.43141494
## 92 -0.3338269018 -0.8670245 -0.82021787 -0.30279450 0.40392161
## 99 -0.1423950448 -0.8670245 -0.78461740 -0.21041342 0.40370254
## 107 -0.7522605641 -0.5224844 -0.14380900 1.12911220 0.42714307
## 110 -0.5935967501 -0.5224844 -0.14380900 1.12911220 0.37850944
## 123 -0.8111955516 -0.8670245 -1.30675758 0.29768250 0.23458021
## 142 -1.1186927669 -0.6373311 0.17066179 1.26768382 0.44061589
## 143 -1.1746358896 -0.5224844 -0.03107419 -1.73470120 0.44061589
## 145 -1.1630958396 -0.5224844 -0.03107419 -1.73470120 0.44061589
## 150 -1.0777089681 -0.5224844 -0.03107419 -1.73470120 -0.05284012
## 158 -0.9107344185 -0.5224844 -0.03107419 -1.73470120 0.07400163
## 166 -0.7178778639 -0.5224844 -0.03107419 -1.73470120 -1.27623862
## 180 -0.4587728745 -0.7521778 -1.27709053 -0.30279450 0.44061589
## 181 -0.5005640019 -0.7521778 -1.27709053 -0.30279450 0.42593818
## 187 -0.2829652003 -0.7521778 -1.27709053 -0.30279450 0.39384437
## 192 1.2749890302 -0.5224844 -0.06074124 -1.50374851 0.36186010
## 194 1.1514202651 -0.9818712 -0.84988492 -1.31898635 0.40194997
## 195 1.1514202651 -0.9818712 -0.84988492 -1.31898635 0.21935482
## 200 1.8323307009 -0.7521778 -0.03700760 -0.67231881 0.44061589
## 213 -0.0676459148 -0.6373311 -0.77868399 0.06672981 0.37533292
## 225 -0.4277144686 -0.1779443 -0.60068166 -0.48755665 0.31081670
## 231 -0.0586703204 -0.1779443 -0.60068166 -0.48755665 0.23742812
## 232 -0.0586703204 -0.1779443 -0.60068166 -0.48755665 0.21322085
## 236 -0.0679783442 -0.1779443 -0.60068166 -0.48755665 0.21990249
## 239 1.1373157596 -0.4076377 -0.64221553 -0.85708096 0.24903885
## 243 1.5388905012 -0.4076377 -0.64221553 -0.85708096 0.17608842
## 246 1.9755128019 -0.2927910 -0.46421320 0.29768250 0.35550706
## 249 1.9145357480 -0.2927910 -0.46421320 0.29768250 0.19755732
## 260 -0.8473828687 -0.5224844 -0.85581834 -2.51994036 0.38617691
## 261 -0.7992280924 -0.5224844 -0.85581834 -2.51994036 0.39570647
## 265 -0.8856597421 -0.5224844 -0.85581834 -2.51994036 0.34192469
## 267 -0.7893501896 -0.5224844 -0.85581834 -2.51994036 0.30008225
## 282 0.6884410603 -0.5224844 -1.14062207 -1.64232012 0.38946296
## 284 0.9925190015 -0.9818712 -1.24742347 -2.24279713 0.42550004
## 291 0.6276539660 -0.6373311 -0.96855315 0.34387304 0.44061589
## 297 1.0283263992 -0.6373311 -0.70748306 -1.13422420 0.39625414
## 305 0.1077818401 -0.2927910 -1.10502160 -0.02565127 0.40534556
## 310 -0.3288879504 -0.6373311 -0.61848189 -0.02565127 0.43338657
## 311 -0.6058016588 -0.6373311 -0.61848189 -0.02565127 -0.06817505
## 317 0.0966691995 -0.6373311 -0.61848189 -0.02565127 0.37270408
## 319 -0.1246813056 -0.6373311 -0.61848189 -0.02565127 0.42210445
## 330 0.6741465952 -0.6373311 0.12912791 -0.71850935 0.20303408
## 333 1.3514003073 -0.9818712 -0.61848189 -0.71850935 0.06107647
## 341 0.4830471675 -0.5224844 -1.09315478 0.80577843 0.44061589
## 343 1.1738829960 -0.9818712 0.08166062 -1.18041474 0.36459848
## 344 0.9199069177 -0.5224844 -0.22687676 -0.39517558 0.44061589
## 347 2.0033893834 -0.7521778 -0.33367816 0.15911089 0.08692678
## 362 -0.7121315839 1.6596029 1.52941294 0.80577843 -0.06598434
## 373 -1.2658165310 1.6596029 1.52941294 0.80577843 -0.09632559
## 379 -1.1440048928 1.6596029 1.52941294 0.80577843 0.44061589
## 392 -0.7727762084 1.6596029 1.52941294 0.80577843 0.23775673
## 394 -0.9516232374 1.6596029 1.52941294 0.80577843 0.44061589
## 402 -1.0547238481 1.6596029 1.52941294 0.80577843 0.44061589
## 403 -1.0239028917 1.6596029 1.52941294 0.80577843 0.21289224
## 405 -1.0389097056 1.6596029 1.52941294 0.80577843 -0.29808940
## 409 -1.1114268095 1.6596029 1.52941294 0.80577843 -0.46042055
## 411 -1.1312301050 1.6596029 1.52941294 0.80577843 -3.87835651
## 413 -1.0643168114 1.6596029 1.52941294 0.80577843 -3.59148386
## 417 -0.9381835908 1.6596029 1.52941294 0.80577843 -3.67056826
## 420 -0.9502935197 1.6596029 1.52941294 0.80577843 -3.37613768
## 421 -0.9194725633 1.6596029 1.52941294 0.80577843 -0.41540159
## 427 -0.8536040479 1.6596029 1.52941294 0.80577843 -3.63683142
## 428 -0.9175729666 1.6596029 1.52941294 0.80577843 -3.70069044
## 430 -0.8675660836 1.6596029 1.52941294 0.80577843 -3.24173801
## 435 -0.7469416934 1.6596029 1.52941294 0.80577843 -2.80458308
## 436 -0.7932443629 1.6596029 1.52941294 0.80577843 -2.70359163
## 448 -0.7584342534 1.6596029 1.52941294 0.80577843 0.34882541
## 450 -0.7646079427 1.6596029 1.52941294 0.80577843 -0.57466575
## 452 -0.6837801032 1.6596029 1.52941294 0.80577843 -0.01516002
## 453 -0.6776064140 1.6596029 1.52941294 0.80577843 0.31125484
## 454 -0.6374774338 1.6596029 1.52941294 0.80577843 0.21026340
## 455 -0.6168668096 1.6596029 1.52941294 0.80577843 -3.83366615
## 458 -0.4824228534 1.6596029 1.52941294 0.80577843 -3.86849834
## 469 -0.4210658801 1.6596029 1.52941294 0.80577843 0.13216481
## 477 -0.7074775720 1.6596029 1.52941294 0.80577843 0.43305797
## 478 -0.8046419430 1.6596029 1.52941294 0.80577843 -0.07879996
## 486 0.0933923952 1.6596029 1.52941294 0.80577843 0.34992076
## 487 -0.1183176566 1.6596029 1.52941294 0.80577843 0.39439205
## 490 -0.9686246278 -0.6373311 1.79641644 0.75958789 -0.13827757
## 498 -0.4285217972 -0.4076377 -0.10227512 0.34387304 0.44061589
## 499 -0.6581830377 -0.4076377 -0.10227512 0.34387304 0.44061589
## 501 -0.6158695213 -0.4076377 -0.10227512 0.34387304 0.44061589
## 502 -0.6251775451 -0.9818712 -0.80241764 1.17530274 0.38683412
## 503 -0.7159307773 -0.9818712 -0.80241764 1.17530274 0.44061589
## 504 -0.7729186782 -0.9818712 -0.80241764 1.17530274 0.44061589
## 506 -0.6126402069 -0.9818712 -0.80241764 1.17530274 0.44061589
## lstat medv crime
## 1 -1.074498970 0.15952779 low
## 5 -1.025486649 1.48603229 low
## 6 -1.042290874 0.67055821 low
## 8 0.909799859 0.49659041 med_low
## 10 0.622727693 -0.39499459 med_low
## 19 -0.134862757 -0.25364575 med_high
## 20 -0.192277190 -0.47110550 med_high
## 21 1.171665689 -0.97126294 med_high
## 33 2.108501199 -1.01475489 med_high
## 43 -0.958269752 0.30087663 med_low
## 44 -0.730012370 0.23563870 med_low
## 48 0.860787538 -0.64507330 med_low
## 52 -0.451342316 -0.22102678 low
## 72 -0.388326475 -0.09055093 med_low
## 84 -0.720209906 0.03992492 low
## 86 -0.857444405 0.44222547 low
## 90 -0.973673624 0.67055821 low
## 92 -0.623585616 -0.05793197 low
## 99 -1.271948607 2.31237936 low
## 107 0.841182610 -0.32975666 med_low
## 110 0.405673128 -0.34062965 med_high
## 123 0.738956911 -0.22102678 med_low
## 142 3.046737061 -0.88427903 med_high
## 143 1.983869868 -0.99300891 med_high
## 145 2.329756820 -1.16697672 med_high
## 150 1.231880827 -0.77554916 med_high
## 158 -1.129112700 2.04055467 med_high
## 166 -0.398128939 0.26825767 med_high
## 180 -1.066096858 1.59476217 low
## 181 -0.713208146 1.87745985 low
## 187 -1.148717628 2.98650460 low
## 192 -1.115109179 0.86627199 low
## 194 -1.067497210 0.93150992 low
## 195 -1.158520092 0.71405016 low
## 200 -1.133313756 1.34468345 low
## 213 0.472890025 -0.01444002 med_low
## 225 -1.192128541 2.42110924 med_high
## 231 -0.140464165 0.19214675 med_high
## 232 -1.036689465 0.99674784 med_high
## 236 -0.248291271 0.15952779 med_high
## 239 -0.881250390 0.12690883 med_low
## 243 -0.200679302 -0.03618599 med_low
## 246 0.813175569 -0.43848654 med_low
## 249 -0.438739148 0.21389273 med_low
## 260 -0.805631380 0.82278004 med_high
## 261 -0.428936683 1.22508059 med_high
## 265 -0.637589136 1.51865126 med_high
## 267 0.299246373 0.88801797 med_high
## 282 -1.129112700 1.39904839 low
## 284 -1.329363040 2.98650460 low
## 291 -1.305557056 0.64881224 low
## 297 -0.737014131 0.49659041 low
## 305 -0.801430324 1.47515931 low
## 310 -0.375723306 -0.24277276 med_high
## 311 -0.001829314 -0.69943824 med_high
## 317 0.794970993 -0.51459745 med_high
## 319 -0.321109577 0.06167090 med_high
## 330 -0.744015891 0.00730596 low
## 333 -0.675398641 -0.34062965 low
## 341 -0.470947244 -0.41674056 low
## 343 -0.560569774 -0.65594629 low
## 344 -0.766421523 0.14865480 low
## 347 0.002371742 -0.57983538 low
## 362 0.215225251 -0.28626471 high
## 373 -0.528361678 2.98650460 high
## 379 1.545559682 -1.02562788 high
## 392 0.855186130 0.07254389 high
## 394 0.352459751 -0.94951696 high
## 402 1.073641047 -1.66713415 high
## 403 1.072240695 -1.13435775 high
## 405 2.062289582 -1.52578531 high
## 409 1.925055083 -0.57983538 high
## 411 -0.356118378 -0.81904111 high
## 413 3.041135653 -0.50372446 high
## 417 1.839633609 -1.63451519 high
## 420 1.412526239 -1.53665830 high
## 421 0.331454470 -0.63420032 high
## 427 0.425278056 -1.34094452 high
## 428 0.261436868 -1.26483361 high
## 430 1.600173411 -1.41705543 high
## 435 0.352459751 -1.17784970 high
## 436 1.486744897 -0.99300891 high
## 448 0.530304459 -1.07999281 high
## 450 0.932205492 -1.03650086 high
## 452 0.710949871 -0.79729513 high
## 453 0.646533677 -0.69943824 high
## 454 0.572315020 -0.51459745 high
## 455 0.848184370 -0.82991410 high
## 458 0.600322060 -0.98213592 high
## 469 0.766963952 -0.37324861 high
## 477 0.843983314 -0.63420032 high
## 478 1.716402630 -1.14523074 high
## 486 -0.290301832 -0.14491587 med_high
## 487 0.325853062 -0.37324861 high
## 490 1.584769539 -1.68888013 med_low
## 498 0.202622083 -0.46023251 med_high
## 499 0.037380543 -0.14491587 med_low
## 501 0.234830180 -0.62332733 med_low
## 502 -0.417733867 -0.01444002 low
## 503 -0.500354637 -0.21015379 low
## 504 -0.982075736 0.14865480 low
## 506 -0.668396881 -1.15610373 low
## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2376238 0.2722772 0.2500000 0.2400990
##
## Group means:
## zn indus chas nox rm
## low 1.14812060 -0.9058553 -0.10828322 -0.9294756 0.3805133
## med_low -0.09023688 -0.3103016 -0.02178633 -0.5706297 -0.1008021
## med_high -0.39214634 0.2448498 0.27340760 0.3799499 0.1658428
## high -0.48724019 1.0172187 -0.02879709 1.0571563 -0.4980867
## age dis rad tax ptratio
## low -0.9599598 0.9984033 -0.6624538 -0.7259014 -0.47889593
## med_low -0.3476450 0.3256522 -0.5475419 -0.4761340 -0.05630499
## med_high 0.4173480 -0.3518439 -0.4110490 -0.2912043 -0.22916404
## high 0.8188552 -0.8600010 1.6371072 1.5133254 0.77958792
## black lstat medv
## low 0.37865777 -0.74298021 0.49002131
## med_low 0.32561889 -0.18387508 0.02667965
## med_high 0.07620643 -0.04591961 0.19666819
## high -0.65000902 0.88140303 -0.66087837
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.091941934 0.82487563 -0.97191792
## indus 0.048657886 -0.17485541 -0.03661729
## chas -0.085216027 -0.05912733 0.02160071
## nox 0.388873747 -0.74182440 -1.33669281
## rm -0.084143860 -0.20165378 -0.19913041
## age 0.283695372 -0.31483509 -0.18636568
## dis 0.007025698 -0.32796666 -0.03576485
## rad 3.162267607 1.10414211 -0.19347746
## tax 0.009950956 -0.19391080 0.80153366
## ptratio 0.129069567 -0.08483473 -0.23779991
## black -0.095773655 0.02211754 0.09997623
## lstat 0.248158882 -0.20383563 0.28350891
## medv 0.228423642 -0.39705787 -0.17994418
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9375 0.0489 0.0137
## [1] low low low med_low med_low med_high med_high
## [8] med_high med_high med_low med_low med_low low med_low
## [15] low low low low low med_low med_high
## [22] med_low med_high med_high med_high med_high med_high med_high
## [29] low low low low low low low
## [36] med_low med_high med_high med_high med_high med_low med_low
## [43] med_low med_low med_high med_high med_high med_high low
## [50] low low low low med_high med_high med_high
## [57] med_high low low low low low low
## [64] high high high high high high high
## [71] high high high high high high high
## [78] high high high high high high high
## [85] high high high high high high high
## [92] high med_high high med_low med_high med_low med_low
## [99] low low low low
## Levels: low med_low med_high high
## predicted
## correct low med_low med_high high
## low 12 16 3 0
## med_low 1 11 4 0
## med_high 0 10 14 1
## high 0 0 0 30
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2662 8.4832 12.6090 13.5488 17.7568 48.8618
While checking for the optimal number of clusters you should look in the graph when defining the k max = 10 for the k value where the curve varies the most dramatically. That appears to be around k = 2
When exploring the pairs using the colors to separate the km clusters, you can see that the variable that better separetes the two clusters (Red Vs Black cluster) would be tax (full-value property-tax rate per $10,000) and rad (index of accessibility to radial highways). An almost complete cluster separation can also be observed when you check the crim (per capita crime rate by town) and black (1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town) variables.
For the reasonable amount of clusters (> 2) I select k = 3 as it was the following position in the curve you could observe a dramatically change.
Now when you check at the pairs with the clusters divided by colors, since we have more clusters it gets a bit more confusing to read the data. Nonetheless, after careful observation you the variables were you can more easily identify the different clusters (Red, Black and Green) are medv (median value of owner-occupied homes in $1000s) and partialy successful will be age (proportion of owner-occupied units built prior to 1940).
I was trying the following approach but I guess I missed something, further work/learning required in this section ; if the peer review knows how to fix this and explain me what is wrong I highly appreciate it!!
k-means clustering km <-kmeans(boston_scaled, centers = 3)
linear discriminant analysis lda.fit <- lda(km ~ . , data = boston_scaled$km)
print the lda.fit object lda.fit
function for the lda biplot arrows lda.arrows <- function(x, myscale = 2, arrow_heads = 0.1, color = “red”, tex = 0.75, choices = c(1,2)){ heads <- coef(x) arrows(x0 = 0, y0 = 0, x1 = myscale * heads[,choices[1]], y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads) text(myscale * heads[,choices], labels = row.names(heads), cex = tex, col=color, pos=3) }
target classes as numeric classes <- as.numeric(boston_scaled$km)
plot the lda results plot(lda.fit, dimen = 2, col = classes, pch = classes) lda.arrows(lda.fit, myscale = 5)
I got stuck in this section as unfortunetely I couldn’t manage to fix the errors in my code
## [1] 404 13
## [1] 13 3
##
## Attaching package: 'plotly'
## The following object is masked from 'package:ggplot2':
##
## last_plot
## The following object is masked from 'package:MASS':
##
## select
## The following object is masked from 'package:stats':
##
## filter
## The following object is masked from 'package:graphics':
##
## layout
## Warning: 'scatter3d' objects don't have these attributes: 'col'
## Valid attributes include:
## 'type', 'visible', 'showlegend', 'legendgroup', 'opacity', 'name', 'uid', 'ids', 'customdata', 'hoverinfo', 'hoverlabel', 'stream', 'x', 'y', 'z', 'text', 'hovertext', 'mode', 'surfaceaxis', 'surfacecolor', 'projection', 'connectgaps', 'line', 'marker', 'textposition', 'textfont', 'error_x', 'error_y', 'error_z', 'scene', 'xcalendar', 'ycalendar', 'zcalendar', 'idssrc', 'customdatasrc', 'hoverinfosrc', 'xsrc', 'ysrc', 'zsrc', 'textsrc', 'hovertextsrc', 'textpositionsrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule'